Method and system of trading a standardized contract

ABSTRACT

A system, method, and corresponding computer program product facilitates trading of a standardized contract. The terms of the contract may be such that it provides a payout from one party to the other based on the price, yield, level or other measure of an asset, basket, index, financial contract, other financial instrument or some economically significant variable observed at or around two specified times, both such times being after the time that the standardized contract is first available for trading. Alternatively, the terms of the contract may be such that it provides that one party has the right but not the obligation either to purchase or to sell some asset at a price which is determined at a first specified time, such right to be exercised at some time not later than the second specified time, both such times being after the time that the standardized contract is first available for trading. These and other related contracts are described.

CROSS-REFERENCE TO RELATED APPLICATIONS

This patent application claims the benefit of priority of and is acontinuation of U.S. patent application Ser. No. 12/697,106 filed onJan. 29, 2010 (pending), which claims the benefit of priority of and isa divisional of U.S. patent application Ser. No. 10/631,107 filed onJul. 31, 2003 (now issued as a U.S. Pat. No. 7,668,774), which claimsthe benefit of priority of Provisional Application No. 60/400,358, filedJul. 31, 2002, and Provisional Application No. 60/429,889, filed Nov.26, 2002. The present application incorporates the disclosure of all ofthese patent applications by reference.

FIELD OF THE INVENTION

This invention relates to a system, method, and corresponding computerprogram product containing code for implementing the method, for tradingand settling a contract. More specifically, this invention relates to asystem, method, and corresponding computer program product for tradingand settling a contract which has a price based on the expectation ofthe rise or fall (measured in some fashion, according to the particularembodiment) of some underlying between two specified times, both suchtimes being after the time that the contract is first available fortrading. The underlying may be an asset, (financial, physical or other),basket, index, contract, other financial instrument or othereconomically significant variable(s) (including weather and geologicaldata).

BACKGROUND OF THE INVENTION

There are a number of terms used herein which whose meanings will bewell known to those of working skill in the field of financial markets.However, for ease of reference, the following table contains definitionsof terms:

Yield—The “yield” of an investment is a measure of its expectedannualised return, (arising from income only or including capitalappreciation or depreciation). In the case of most interest rate futurescontracts, the “yield” which corresponds to the contract's price is 100%minus the price.

Options—The collective term for “call options” and “put options” (q.v.),and occasionally also for other “derivatives” (q.v.).

Call option or Call—A contract which, if “physically-settled,” gives oneparty, (the “option buyer”), the right, but not the obligation, topurchase from the other party, (the “option seller”), a certain quantityof an asset or other financial instrument at a fixed price, (the “strikeprice” or “strike”), at a specified time or times. If the call option is“cash-settled,” then, upon electing to exercise the option, the optionbuyer receives from the option seller a cash amount equal to theeconomic benefit that would arise if the option had beenphysically-settled, i.e. by reference to the excess of the underlyingover the strike.

Put option or Put—A contract which, if “physically-settled,” gives oneparty, (the “option buyer”), the right, but not the obligation, to sellto the other party, (the “option seller”), a certain quantity of anasset or other financial instrument at a fixed price, (the “strikeprice” or “strike”), at a specified time or times. If the put option is“cash-settled,” then, upon electing to exercise the option, the optionbuyer receives from the option seller a cash amount equal to theeconomic benefit that would arise if the option had beenphysically-settled, i.e. by reference to the excess of the strike overthe underlying.

Derivatives—Contracts (including options) whose economic performance aredependent on the evolution of the price, yield or level of an asset,basket, index, contract or other financial instrument, or on theevolution of some other economically significant variable (includingweather and geological data).

Underlying—The asset, basket, index, contract, other financialinstrument or other economically significant variable, (and,interchangeably, the price, yield or level thereof), to which aderivative contract's performance is linked.

Hedge—“To hedge” is to invest in an asset or other financial instrumentor to enter into a financial contract, (in any instance, a “hedge”), soas to offset the risk associated with other assets, financialinstruments and/or financial contracts.

Delta-hedge—“To delta-hedge” an option position is to manage dynamicallythe amount, (“the delta-hedge”), of a direct exposure to the underlyingof that option so as to neutralise the option's instantaneous pricesensitivity to small changes in the price, yield or level of theunderlying.

Mark-to-market revaluation—The revaluation of a position based on itsprevailing market price.

Bid price—A price proffered for the purchase of an asset or otherfinancial instrument.

Offered price—A price proffered for the sale of an asset or otherfinancial instrument.

Mid-market—The price midway between the best readily-available bid andoffered prices.

Expiration—The last time, according to the terms of an option contract,that the option buyer can exercise his right.

Liquidity—The ease with which one can buy or sell an asset or otherfinancial instrument quickly and in large volume without substantiallyaffecting the asset or financial instrument's price, usuallycharacterised by narrowly separated bid and offered prices.

Term structure—The observed dependence of a financial measure on term ormaturity, (e.g. the dependence of yield on redemption date in the caseof government bonds, or the dependence of implied volatility on theexpiration date in the case of options, etc.)

Duration—A measure of the sensitivity to interest rates of the value ofa portfolio of bonds or other interest-rate sensitive instruments. It isdefined as that maturity of notional zero-coupon bond having the samemonetary value as the portfolio which also would have the samesensitivity as the portfolio to a single small change in interest ratesapplied equally to all maturities. Duration therefore takes no accountof non-uniform changes in the interest rate term structure.

Present-Value-of-a-Basis-Point—A measure of the sensitivity of aportfolio of bonds or other interest-rate sensitive instruments inmonetary terms to a 0.01% change in interest rates applied to allmaturities. Present-Value-of-a-Basis-Point therefore takes no account ofnon-uniform changes in the interest rate term structure.

Futures (and options) exchange—An organisation that brings togetherbuyers and sellers of futures (and options) contracts by, for example,open outcry or electronic trading.

Strike price or Strike—See “Call option or Call” and “Put option orPut”.

Clearinghouse—The clearinghouse of a futures (and options) exchange actsas the seller to all buyers and the buyer to all sellers of futures (andoptions) contracts transacted. As participants may both buy and sellcontracts on many occasions, this function of the clearinghouseeliminates the need for keeping track of the complex and long list ofsuccessive buyers and sellers of each contract. Additionally,participants are not exposed to default on the contracts by the otherparticipants. Each participant holds an account with a “clearinghousemember” which margins the account of the participant. The clearinghousemargins the accounts of the clearinghouse members.

Margin—“Margin” is the amount of money that the clearinghouse orclearinghouse member requires as deposit in order to maintain aposition. “Margining” is the practice, usually undertaken daily, ofmaintaining a minimum margin with a clearinghouse or clearinghousemember taking account of accrued profits and losses on participants'positions. The purpose of margining is to protect the clearinghouse andclearinghouse members from defaults.

Cash-settled—See “Call option or Call” and “Put option or Put”

At-the-money—An option is at-the-money when the price of the underlyingand the strike are the same.

Discount factor—A multiplier used to convert a future cashflow to itspresent value, and is dependent on interest rates and the period untilthat cashflow occurs.

Premium—The cost of purchasing an option.

Basis risk—The residual risk arising from the use of a proxy hedge.

Rolling a contract—Closing a position in one futures or options contractand simultaneously establishing an identical position in the futures oroptions contract with a later expiration.

Contract month—The month in which a futures or options contract expires.Usually all contracts of the same type but with different expirationsexpire in different months, and so the contract month is used todistinguish them.

Proxy hedge—A hedge whose performance is not perfectly correlated withthe performance of the asset, basket, index, contract, other financialinstrument or other economically significant variable being hedged.

Straddle—An option position comprising a call option and a put option(q.v.) with identical underlyings, quantities, strikes and expirations.

Strangle—An option position comprising a call option and a put option(q.v.) with identical underlyings, quantities and expirations, butdifferent strikes.

Physically-settled option—See “Call Option or Call” and “Put option orPut”.

Over-the-counter—An “over-the-counter” (OTC) market is an informalmarket that does not involve a futures exchange.

Open outcry—The method of trading futures and options contracts wherebybrokers (and traders) congregate at the designated exchange premises andexpress their intentions to buy or sell by calling out and by the use ofhand signals.

Electronic trading—The method of trading whereby brokers' and traders'orders are submitted to a computer system which identifies and executesmatching trades.

In-the-money—An option is in-the-money if, in the case of a call option,the underlying exceeds the strike or, in the case of a put option, thestrike exceeds the underlying.

Out-of-the-money—An option is out-of-the-money if, in the case of a calloption, the strike exceeds the underlying or, in the case of a putoption, the underlying exceeds the strike.

The pricing and hedging of options and certain other derivatives, (whichwill be collectively referred to as “options” herein for the sake ofconvenience), was given a strong mathematical foundation by the work ofF. Black and M. Scholes. See, e.g., Black, F. and Scholes, M., (1973),“The Pricing of Options and Corporate Liabilities,” Journal of PoliticalEconomy, 81, 637. Black and Scholes made a number of idealisedassumptions about markets and price behaviour. In particular, theyassumed that the price, S, of the underlying asset of an option followsa geometric Brownian process governed by the stochastic differentialequation:

$\frac{\mathbb{d}S}{S} = {{\mu\;{\mathbb{d}t}} + {\sigma{\mathbb{d}z}}}$where μ is a drift rate, σ is a diffusion constant known as thevolatility, dt is an infinitesimal increment in time, and dz is theinfinitesimal change in a variable, z, which follows a standard Wienerprocess, (where dz has expectation equal to zero and variance equal todt).

From this model, Black and Scholes, and many others have been able toderive formulae and algorithms for valuing a wide variety of optiontypes. Somewhat paradoxically, the option values do not depend on thedrift rate, μ. The central problem for obtaining numerical values foroption prices has thus been the determination of the volatility, a,since this is not directly observable.

Although Black and Scholes' model assumes that the volatility of theunderlying is constant, it is observed that volatilities in general areneither constant nor entirely predictable.

Future volatility can either be estimated from previously experiencedvolatility, (known as the “historical volatility”), or by deducing thevolatility consistent with the prevailing observable prices of options,(known as the “implied volatility”). Because the latter isforward-looking, it is generally regarded as the better estimate.

The implied volatility at which options trade for many underlyings, (orthe mid-market implied volatility between bid and offered prices),usually varies from maturity to maturity so that there is a discernibleterm structure of implied volatility. Financial institutions frequentlymodel this by assuming that the volatility of the underlying variesthrough time in an entirely deterministic way, (i.e. they assumeσ=σ(t)), which we refer to as the “time-dependent Black-Scholesframework”.

Because of the enormous growth in the volume of options traded byfinancial institutions, (as well as by end users), financialinstitutions have become very sensitised to their exposure to changes inboth (i) future realised volatility—which affects the cost or benefit ofdelta-hedging their option positions—and (ii) implied volatility—largelydriven by supply and demand factors and which affects the mark-to-marketrevaluation of their option positions.

The sensitivity of the value of an option position, X, to small changesin the overall level of implied volatility is usually referred to as the“vega,” κ, of the option position, defined as:

$\kappa = \frac{\partial X}{\partial\sigma}$

However, financial institutions with portfolios containing many optionpositions are not just exposed to uniform changes in volatility, butalso are exposed to non-uniform changes in the implied volatility termstructure. Just as the Duration or the Present-Value-of-a-Basis-Point ofa bond portfolio is no longer considered a sufficiently accurate measureof interest rate exposure, the overall vega of an option portfolio is nolonger considered a sufficiently accurate measure of volatilityexposure.

In the case of interest rates, financial institutions are able to hedgetheir exposure to changes in the term structure, (at least, in respectof the major currencies), very efficiently by trading the highly liquidinterest rate futures and bond futures on the futures exchanges. Byselecting the appropriate contract or contracts, exposure to interestrates at a specific maturity can be hedged.

In the case of volatility, however, the ability of financialinstitutions to hedge their exposure to changes in the term structure onthe futures exchanges is much more restricted. Whilst regularexchange-traded call and put options may be used to hedge volatilityexposure, they suffer from a number of disadvantages:

-   -   1) The liquidity of exchange-traded options, (except at short        maturities), is typically relatively low. This can be attributed        to the fact that, for each maturity, many different option        strikes are available, which has the effect of fragmenting        liquidity.    -   2) The price, (and hence the profitability), of regular call and        put options are sensitive to a number of other variables besides        volatility, not the least of which is the level of the        underlying.    -   3) The vega of regular call and put options is not constant and        is itself dependent on a number of other variables, particularly        the level of the underlying. This means that, unless the terms        of the calls or puts traded match quite closely the options        being hedged, the net exposure to volatility may behave quite        unpredictably over time.

Points 2 and 3 are also important considerations for financialinstitutions or speculators who wish to take positions that express apure volatility view. If the volatility exposure of a position iscontaminated by a number of other exposures, then the profitability ofthat position will be contaminated by the profitability (or otherwise)relating to those other exposures.

It would be desirable to provide a contract which enables speculatorsand financial institutions to take a view on volatility uncontaminatedby other exposures. It also would be desirable to provide a contractwhich enables financial institutions to hedge volatility exposureuncontaminated by other exposures.

Aside from being exposed to changes in the term structure of volatility,market professionals are also exposed to sudden jumps in the underlying.Sudden jumps in the underlying can cause a material mismatch between theperformance of the delta-hedge and the performance of the optionportfolio being hedged. This risk is usually measured by the “gamma,” Γ,of the portfolio, which is the local sensitivity of the “delta,” (thetheoretical quantity of delta-hedge required to hedge the portfolio), tochanges in the underlying. The delta is:

$\Delta = \frac{\partial X}{\partial S}$and therefore the gamma is:

$\Gamma = {\frac{\partial\Delta}{\partial S} = \frac{\partial^{2}X}{\partial S^{2}}}$

It would be desirable if the contract that is provided also enablesfinancial institutions to hedge their gamma exposures.

Other relevant background is provided in F. Black, “The Pricing ofCommodity Contracts,” (1976), Journal of Financial Economics, 3, 167;and D. R. Cox and H. D. Miller, “The Theory of Stochastic Processes,”(1965), London: Chapman & Hall.

SUMMARY OF THE INVENTION

In view of the foregoing, according to one feature of the presentinvention, financial institutions, through certain forms of contractdescribed herein, can trade and hedge the volatility level of someunderlying relating to a future time slice. If there is a series of suchcontracts available relating to contiguous (or nearly contiguous) timeslices, then it will be possible to combine positions in order to createa chosen time profile of volatility exposure.

The terms of the contract can be described very simply without referenceto volatility at all. This means that the contract terms can be easilyunderstood by speculators who have no technical knowledge of volatility;an assessment of cheapness or richness can be made based simply on acomparison of break-even and historical performance.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention now will be described in detail with reference toexemplary embodiments. In the drawings:

FIG. 1 is an exemplary flow of the overall process according to theinvention;

FIGS. 2 a and 2 b describe exemplary flow of transactions relative tooptions traded on a futures and options exchange;

FIG. 3 a illustrates calculation of an option price at a given point intime;

FIG. 3 b illustrates calculation of the price of a call-type option at agiven point in time; and

FIG. 3 c illustrates calculation of a put-type option at a given pointin time.

DETAILED DESCRIPTION OF EMBODIMENTS

In the embodiments described below, the contract ideally trades throughan exchange and settles through a clearinghouse, thereby providingparticipants with the benefits of anonymity and much reduced likelihoodof counterparty default on the contracts. Additionally, theparticipants' positions are margined to protect the clearinghousemembers from default on the part of the participants.

In accordance with one embodiment, the contract has a payout equal tothe percentage gain, if any, of some underlying from a First ReferenceTime to a Second Reference Time (later than the First Reference Time)multiplied by some specified currency amount. The time interval betweenthe First Reference Time and the Second Reference Time is the period,the “Reference Period,” in relation to which a volatility exposure maybe gained through the contract and might typically be three months.Three months would generally be considered neither too long as to be tooblunt a hedge for short periods of volatility, nor too short so as torequire a combination of too large a number of different contracts (withcontiguous Reference Periods) to hedge the volatility relating to aperiod of, say, a few years. The contract is first made available fortrading prior to the First Reference Time.

Effectively, this contract is a cash-settled call option contract whosestrike is fixed at-the-money at the First Reference Time.

To illustrate how this contract provides an exposure to volatility, wecan consider the case in which the underlying is the price of a futurescontract or the corresponding yield of an interest rate futurescontract. In that case, at any time prior to and until the FirstReference Time, the price of the option contract is related to theimplied volatility for the Reference Period, in the Black-Scholesframework, by the equation:

$V_{C}^{-} = {D \cdot \left\lbrack {{N\left( \frac{\sigma\sqrt{\tau}}{2} \right)} - {N\left( {- \frac{\sigma\sqrt{\tau}}{2}} \right)}} \right\rbrack}$

where:

-   -   V_(C) ⁻ is the price of the option contract per unit of the        specified currency amount, at a time prior to and until the        First Reference Time, (the ‘−’ superscript denotes ‘prior to and        until the First Reference Time’ and the ‘C’ subscript denotes a        ‘call’-type contract);    -   D is the applicable discount factor representing the cost of        financing the premium from the time of payment until the final        settlement of outstanding contracts after the Second Reference        Time;    -   N(·) is the standard cumulative Normal distribution function        given by:

${N(z)} = {\int_{- \infty}^{z}{\frac{{\mathbb{e}}^{{- x^{2}}/2}}{\sqrt{2\pi}}\ {\mathbb{d}x}}}$

-   -   σ is the prevailing implied volatility of the underlying        relating to the Reference Period; and    -   τ is the length of the Reference Period in years.

The price of the option contract therefore reflects the prevailingmarket level of the implied volatility of the underlying relating to theReference Period.

Furthermore, if the option contract's terms specify that the payment ofthe premium is deferred until maturity so that, (as with a standardcash-settled futures contract), there is simply a net settlement of thetraded price against the final settlement price, then D is 1, (i.e. nofinancing), and the option contract's price is not dependent on D.Otherwise, the exposure to changes in D can be mitigated by financing ordepositing the premium at prevailing rates to the Second Reference Time.For simplicity of exposition, we consider the specific case for which Dis 1, although the extension to the case where D≠1 is straightforward.

In the case where D is 1, we see that the price of the option contract,V_(C) ⁻, prior to the First Reference Time, is a function of the impliedvolatility only, (as τ is fixed). Moreover, the relationship betweenV_(C) ⁻ and σ is very nearly a proportionate one for typical values of σand √τ, (i.e. σ√τ<1). This is because the Maclaurin series for V_(C) ⁻(in terms of σ√τ) has no zero-th order or second order term −V_(C) ⁻ isthe difference between N(z) and N(−z) in the near-linear part of N(z)either side of the inflection. More explicitly, V_(C) ⁻ is approximatelygiven by:

$V_{C}^{-} \approx \frac{\sigma\sqrt{\tau}}{\sqrt{2\;\pi}}$

At a volatility of 50% and for τ=0.25, the Black-Scholes value of theoption contract is 9.9477% whilst the linear approximation gives9.9736%, overstating the Black-Scholes value by only 0.0259%, (allvalues given to the fourth decimal place when expressed as a decimal).(As √τ=0.50, it may be considered convenient to use 0.40 as anapproximate value of 1/√(2π), which very readily gives a value of10.0000%).

Prior to the First Reference Time, the option contract's vega istherefore very nearly constant, and is given by:

$\frac{\partial V_{C}^{-}}{\partial\sigma} = {{{\exp\left( {- \frac{\sigma^{2}\tau}{8}} \right)} \cdot \frac{\sqrt{\tau}}{\sqrt{2\pi}}} \approx \frac{\sqrt{\tau}}{2\pi}}$

Whilst the relationship between the option contract's price andvolatility is not perfectly linear, the non-linearity is a small priceto pay for the benefit of simplicity, and, in any case, is somethingwhich market professionals should certainly be able to cope with.

At the First Reference Time, the option contract will be struckat-the-money and its market price will determine the implied volatilityof the then at-the-money option expiring at the Second ReferenceTime—the option contract automatically converges to the impliedvolatility of the relevant at-the-money option at the First ReferenceTime. Traders can, as a result of this convergence, confidently use thiscontract to speculate on the implied volatility relating to theReference Period up until the First Reference Time.

From the First Reference Time the option contract takes the form of aregular call option with a known strike, and at any time thereafter andprior to the Second Reference Time, the price of the option contract isrelated to the implied volatility for the remaining period, in theBlack-Scholes framework, by the equation, (more generally allowing D≠1):

$V_{C}^{+} = {D \cdot \left\lbrack {{\frac{F}{K} \cdot {N\left( \frac{{\ln\left( {F/K} \right)} + {{1/2}\sigma_{0}^{2}\theta}}{\sigma_{0}\sqrt{\theta}} \right)}} - {N\left( \frac{{\ln\left( {F/K} \right)} - {{1/2}\;\sigma_{0}^{2}\theta}}{\sigma_{0}\sqrt{\theta}} \right)}} \right\rbrack}$

where:

-   -   V_(C) ⁺ is the price of the option contract per unit of the        specified currency amount, at a time after the First Reference        Time and prior to the Second Reference Time, (the ‘+’        superscript denotes ‘after the First Reference Time and prior to        the Second Reference Time’);    -   F is the prevailing price of the underlying futures contract or        the corresponding yield of the underlying interest rate futures        contract;    -   K is the strike price, the level of the underlying determined at        (or around) the First Reference Time;    -   σ₀ is the prevailing implied volatility relating to the        remaining time until the Second Reference Time; and    -   θ the remaining time in years until the Second Reference Time.

Once the strike has been fixed, the option contract's price also becomessensitive to the underlying. This may be dealt with in the normalfashion by delta-hedging, (as it would typically be with all otheroption positions). Below we demonstrate that, except in certaincircumstances, the number of these contracts, (or of contracts thatcomprise other embodiments described below), which hedges an optionposition against changes in the implied volatility relating to theremaining Reference Period is the same as the number which hedges thegamma of that option, and so the changes in the delta of that number ofcontracts tend to track the changes in delta of the option being hedged.

As an alternative to delta-hedging the option contracts, the buyer (orseller) may be prepared to run a basis risk by rolling out of theposition in the option contract prior to the determination of its strikeinto the next-nearest contract month thereby using the latter contractas a proxy hedge for the volatility exposure relating to the ReferencePeriod of the former contract—in a similar way that the next interestrate future is often used as a proxy hedge for ‘spot’ (i.e. near-term)interest rate exposure.

Once the Second Reference Time has occurred, any outstanding contractsare settled at their settlement price defined as:

$V_{C}^{f} = {{Max}\left\lbrack {\frac{F^{f}}{K} - \text{1,0}} \right\rbrack}$

where:

-   -   V_(C) ^(f) is the final settlement price of the option contract        per unit of the specified currency amount, (the ‘f’ superscript        denotes ‘final’); and    -   F^(f) is the price of the underlying futures contract or the        corresponding yield of the underlying interest rate futures        contract as determined at (or around) the Second Reference Time.

In accordance with another embodiment, a contract has a payout equal tothe percentage fall, if any, of the underlying from the First ReferenceTime to the Second Reference Time multiplied by some specified currencyamount.

Effectively, this contract is a cash-settled put option contract whosestrike is fixed at-the-money at the First Reference Time.

Again, we consider the case where the underlying is the price of afutures contract or the corresponding yield of an interest rate futurescontract. At any time prior to and until the First Reference Time, theprice of the option contract is related to the implied volatility of theReference Period, in the Black-Scholes framework, in precisely the sameway as for the corresponding call option contract, (as given above),i.e.

$V_{P}^{-} = {{D \cdot \left\lbrack {{N\left( \frac{\sigma\sqrt{\tau}}{2} \right)} - {N\left( {- \frac{\sigma\sqrt{\tau}}{2}} \right)}} \right\rbrack} = V_{C}^{-}}$the ‘P’ subscript denoting a ‘put’-type contract. For these types ofunderlying, prior to the fixing of the strike, the price of the call andthe put option contracts should theoretically track each otherperfectly.

From the First Reference Time the option contract takes the form of aregular put option with a known strike, and at any time thereafter andprior to the Second Reference Time, the price of the option contract isrelated to the implied volatility for the remaining period, in theBlack-Scholes framework, by the equation:

$V_{P}^{+} = {{D \cdot \left\lbrack {{N\left( \frac{{\ln\left( {K/F} \right)} + {{1/2}\sigma_{0}^{2}\theta}}{\sigma_{0}\sqrt{\theta}} \right)} - {\frac{F}{K}{N\left( \frac{{\ln\left( {K/F} \right)} - {{1/2}\sigma_{0}^{2}\theta}}{\sigma_{0}\sqrt{\theta}} \right)}}} \right\rbrack} = {V_{C}^{+} + {D \cdot \left\lbrack {1 - \frac{F}{K}} \right\rbrack}}}$

Because the strike is the same for both the call and the put optioncontracts, their prices continue to be linked after the First ReferenceTime through the formula above, (known as the put-call parityrelationship). It may therefore be argued that there is a theoreticalredundancy in having both calls and puts. However, there is a practicalbenefit. The existence of both allows speculators to establish prior tothe First Reference Time one of a variety of ‘static strategies’ atinception by mixing calls and puts in differing proportions. (Staticstrategies are strategies which are evaluated purely on their payoutprofile if held to expiration). If, however, only one of calls, puts or(indeed) straddles were made available, then speculators would need totrade in the underlying close to the time that the strike is set inorder to bias their position so as to obtain the strategy they desire.

Once the Second Reference Time has occurred, any outstanding put optioncontracts are settled at their settlement price defined as:

$V_{P}^{f} = {{Max}\left\lbrack {{1 - \frac{F^{f}}{K}},0} \right\rbrack}$

In the case where the underlying is the level of an equity index, forexample, then the price of the call and the put option contract willadditionally depend on the cost of carry of the shares comprising theindex over the Reference Period. If this cost of carry is positive, wewould expect, at any time prior to and until the First Reference Time,the price of the call option to be greater than the price of the putoption, and vice versa. The price of the call option during this period,in this case, can be expressed as, (D=1):

$V_{C}^{-} = {{R \cdot {N\left( {\frac{\sigma\sqrt{\tau}}{2} + ɛ} \right)}} - {N\left( {{- \frac{\sigma\sqrt{\tau}}{2}} + ɛ} \right)}}$where:

$ɛ = \frac{\ln(R)}{\sigma\sqrt{\tau}}$and R is the “carry factor”, i.e. the anticipated ratio of the forwardlevel to the spot level of the index as at the First Reference Time.Similarly, the price of the put option prior to and until the FirstReference Time, in this case, can be expressed as, (D=1):

$V_{P}^{-} = {{N\left( {\frac{\sigma\sqrt{\tau}}{2} - ɛ} \right)} - {R \cdot {N\left( {{- \frac{\sigma\sqrt{\tau}}{2}} - ɛ} \right)}}}$

For a three month Reference Period, R will typically be in the range of0.98 to 1.02. The price-volatility graphs for the call and the put arestill virtually linear except at exceptionally low volatilities, (i.e.σ<10%), and (as before) at exceptionally high volatilities.

Whilst a position in such a call option or put option prior to the FirstReference Time creates an exposure to the carry factor, an equalposition in both is virtually insensitive to the carry factor (attypically volatility and carry factor levels) and hence provides avirtually pure exposure to the volatility. Conversely, an equal andopposite position in both provides a pure exposure to the carry factor.

For some underlyings, interest rates perhaps, it may be more appropriatethat the call and put option contracts have payouts which are theabsolute rise or fall, if any, in that underlying, (rather than thepercentage rise or fall), multiplied by some specified currency amount,(e.g. a change in interest rates from 4% to 5% and from 5% to 6% areboth considered as a change of 1% rather than a change of 25% and 20%respectively). These embodiments also are within the contemplation ofthe invention.

If the underlying in these embodiments is the price of a futurescontract or the corresponding yield of an interest rate futurescontract, then at any time prior to and until the First Reference Time,the price of both the call and the put option contract is related to theimplied volatility for the Reference Period, in the Black-Scholesframework, by the slightly different equation:

$V_{C,P}^{-} = {D \cdot F \cdot \left\lbrack {{N\left( \frac{\sigma\sqrt{\tau}}{2} \right)} - {N\left( {- \frac{\sigma\sqrt{\tau}}{2}} \right)}} \right\rbrack}$

We see here that, in this case, the price of the option contract priorto the First Reference Time depends on the level of the underlying, butin a relatively simple fashion which may be delta-hedged:

$\frac{\partial V_{C,P}^{-}}{\partial F} = {{D \cdot \left\lbrack {{N\left( \frac{\sigma\sqrt{\tau}}{2} \right)} - {N\left( {- \frac{\sigma\sqrt{\tau}}{2}} \right)}} \right\rbrack} = \frac{V_{C,P}^{-}}{F}}$

The vega of the option contract is given by:

$\frac{\partial V_{C,P}^{-}}{\partial\sigma} = {{D \cdot F \cdot {\exp\left( {- \frac{\sigma^{2}\tau}{8}} \right)} \cdot \frac{\sqrt{\tau}}{\sqrt{2\pi}}} \approx {D \cdot F \cdot \frac{\sqrt{\tau}}{\sqrt{2\pi}}}}$and so is proportional to the level of the underlying. If F tends tomake small fluctuations in percentage terms, then the vega of the optioncontract will similarly make small percentage fluctuations. If F hasdiffused by a material amount, some adjustment will be necessary torestore the original desired vega.

From the First Reference Time, the option contract again takes the formof a regular call or put option with a known strike, and at any timethereafter and prior to the Second Reference Time, the prices of theoption contracts are related to the implied volatility for the remainingperiod, in the Black-Scholes framework, by the equations (in this case):

$V_{C}^{+} = {D \cdot \left\lbrack {{F \cdot {N\left( \frac{{\ln\left( {F/K} \right)} + {\frac{1}{2}\sigma_{0}^{2}\theta}}{\sigma_{0}\sqrt{\theta}} \right)}} - {K \cdot {N\left( \frac{{\ln\left( {F/K} \right)} - {\frac{1}{2}\sigma_{0}^{2}\theta}}{\sigma_{0}\sqrt{\theta}} \right)}}} \right\rbrack}$

$V_{P}^{+} = {{D \cdot \left\lbrack {{K \cdot {N\left( \frac{{\ln\left( {K/F} \right)} + {\frac{1}{2}\sigma_{0}^{2}\theta}}{\sigma_{0}\sqrt{\theta}} \right)}} - {F \cdot {N\left( \frac{{\ln\left( {K/F} \right)} - {\frac{1}{2}\sigma_{0}^{2}\theta}}{\sigma_{0}\sqrt{\theta}} \right)}}} \right\rbrack} = {V_{C}^{+} + {D \cdot \left\lbrack {K - F} \right\rbrack}}}$

Once the Second Reference Time has occurred, any outstanding contractsare settled at their settlement prices defined (in this case) as:

$\begin{matrix}{V_{C}^{f} = {{Max}\left\lfloor {{F^{f} - K},0} \right\rfloor}} \\{V_{P}^{f} = {{Max}\left\lfloor {{K - F^{f}},0} \right\rfloor}}\end{matrix}$

Another embodiment comprises a contract which is a physically-settledcall option in favour of one party, (the buyer), expiring at the SecondReference Time. The strike for this call option contract is determinedat the First Reference Time. The quantity of the underlying asset whichthe buyer has the right to purchase may be, in two differentembodiments, either specified in absolute terms or in value terms byreference to its price which is determined at the First Reference Time.

Another embodiment comprises a contract which is a physically-settledput option in favour of one party, (the buyer), expiring at the SecondReference Time. The strike for this put option contract is determined atthe First Reference Time. The quantity of the underlying asset which thebuyer has the right to deliver may be, in two different embodiments,either specified in absolute terms or in value terms by reference to itsprice which is determined at the First Reference Time.

Three other embodiments comprise (i) a contract, which shall be calledherein a “relative quadratic call,” which has a payout equal to thesquare of the percentage gain, if any, of the underlying from the FirstReference Time to the Second Reference Time multiplied by some specifiedcurrency amount, (ii) a contract, which shall be called herein a“relative quadratic put,” which has a payout equal to the square of thepercentage fall, if any, of the underlying from the First Reference Timeto the Second Reference Time multiplied by some specified currencyamount, and (iii) a contract, which shall be called herein a “relativequadratic straddle,” which has a payout equal to the square of thepercentage gain or fall of the underlying from the First Reference Timeto the Second Reference Time multiplied by some specified currencyamount.

In the case where the underlying is the price of a futures contract orthe corresponding yield of an interest rate futures contract, then atany time prior to and until the First Reference Time, the price of therelative quadratic straddle and the implied volatility for the ReferencePeriod are related, in the Black-Scholes framework, by the equation:

Q_(S)⁻ = D ⋅ (𝕖^(σ²τ) − 1) ≈ D ⋅ σ²τ

where:

-   -   Q_(S) ⁻ is the price of the relative quadratic straddle per unit        of the specified currency amount, at a time prior to and until        the First Reference Time, (the ‘S’ subscript denotes a        ‘straddle’-type contract).

In those circumstances where D is 1, the price of the contract prior tothe fixing of the strike is a function of the implied volatility only.In this case, the value of the contract is very closely proportional tothe square of the volatility for typical values of σ and τ.

At any time from the First Reference Time and prior to the SecondReference Time the price of the contract and the implied volatility forthe remaining period are related, in the Black-Scholes framework, by theequation:

$Q_{S}^{+} = {D \cdot \left\lbrack {\left( {\frac{F}{K} - 1} \right)^{2} + {\left( \frac{F}{K} \right)^{2}\left( {{\mathbb{e}}^{\sigma_{0}^{2}\theta} - 1} \right)}} \right\rbrack}$

where:

-   -   Q_(S) ⁺ is the price of the quadratic straddle per unit of the        specified currency amount of the contract, at some time during        the Reference Period.

The relative quadratic straddle has the feature that, once its strikehas been set, its gamma is independent of the level of the underlying:

$\Gamma = {{{D \cdot \frac{2}{K^{2}}}{\mathbb{e}}^{\sigma_{0}^{2}\theta}} \approx {{D \cdot \frac{2}{K^{2}}}\left( {1 + {\sigma_{0}^{2}\theta}} \right)}}$

This is to be compared with the gammas of the linear contracts mentionedpreviously which vary considerably with the level of the underlying.Additionally, the gamma of the relative quadratic straddle issignificant right up to the Second Reference Time. These features of therelative quadratic straddle may make them more appealing for hedging thegamma of option positions, and we return to this point later.

Once the Second Reference Time has occurred, any outstanding contractswill be settled at their settlement prices defined as:

$Q_{S}^{f} = \left( {\frac{F}{K} - 1} \right)^{2}$

Three further embodiments comprise (i) a contract, which shall be calledherein an “absolute quadratic call,” which has a payout equal to thesquare of the absolute gain, if any, of the underlying from the FirstReference Time to the Second Reference Time multiplied by some specifiedcurrency amount, (ii) a contract, which shall be called herein an“absolute quadratic put,” which has a payout equal to the square of theabsolute fall, if any, of the underlying from the First Reference Timeto the Second Reference Time multiplied by some specified currencyamount, and (iii) a contract, which shall be called herein an “absolutequadratic straddle,” which has a payout equal to the square of theabsolute gain or fall of the underlying from the First Reference Time tothe Second Reference Time multiplied by some specified currency amount.

Further embodiments comprise contracts as described with respect to oneof the embodiments above but whose strike is determined by applying someprescribed adjustment (absolute and/or multiplicative) to the price ofthe underlying as determined at the First Reference Time. Thesecontracts become in-the-money or out-of-the-money call or put options atthe First Reference Time. These contracts allow speculators andfinancial institutions the ability to trade or hedge the forward “skew,”(the observed dependence of volatility on strike price).

Further embodiments comprise a combination of a call option and putoption contract as described with respect to one of the embodimentsabove to form a straddle or strangle.

Other embodiments comprise contracts whose payout varies in some otherfashion as the level of the underlying determined at (or around) theSecond Reference Time varies relative to the level determined at (oraround) the First Reference Time, in accordance with some otherfunction.

We now turn our attention to how these contracts may be used to hedgethe gamma as well as the vega of an options portfolio.

The gamma of the contracts whose embodiments have been described aboveis non-zero during the Reference Period. “Near-term contracts,” (beingthose in their Reference Period), could therefore be used to hedge thegamma exposure of an option portfolio.

We now demonstrate that within the time-dependent Black-Scholesframework, with certain limited exceptions, the number of near-termcontracts which hedges the gamma of an option is precisely the samenumber which hedges that option to small changes in the volatilitypertaining to the remaining period up to the contract's Second ReferenceTime.

The value of an option at some time, t₀, can be expressed as thediscounted expectation of its value at a later time, t₀+Δt, under theassumption of “risk-neutrality”. According to the assumption ofrisk-neutrality, the drift rate of the underlying is changed from itstrue value, μ, so that the underlying is assumed to follow the modifiedstochastic differential equation:

$\frac{\mathbb{d}S}{S} = {{{\mu^{\prime}(t)}{\mathbb{d}t}} + {{\sigma(t)}{\mathbb{d}z}}}$where μ′(t) is the instantaneous rate for the cost of carry of theunderlying. The risk-neutral probability distribution function forln(S(t₀+Δt)) is Normal with an adjusted mean equal to:

${\ln\; S_{0}} + {\mu_{0}^{\prime}\Delta\; t} - {\frac{1}{2}\sigma_{0}^{2}\Delta\; t}$and (unadjusted) variance equal to:

σ₀²Δ twhere S₀=S(t₀), and μ₀′ and σhd 0 ² are the mean values of it μ′(t) andσ²(t) respectively over the interval t₀ to t₀+Δt.

Now, provided that the value of the option at time t₀+Δt can bedetermined by reference to the level of S at that time and does notdepend on any value of S between t₀ and t₀+Δt, (which therefore excludesoptions which are capable of being exercised between t₀ and t₀+Δt), thenthe value of the option at time t₀ is:

${X\left( {S_{0},t_{0}} \right)} = {{PV}\left\lbrack {\int_{- \infty}^{\infty}{{\frac{{\mathbb{e}}^{{- {({{\ln\; S} - {({{lnS}_{0} + {\mu_{0}^{\prime}\Delta\; t} - {\frac{1}{2}\sigma_{0}^{2}\Delta\; t}})}})}^{2}}/{({2\sigma_{0}^{2}\Delta\; t})}}}{\sqrt{2{\pi\sigma}_{0}^{2}\Delta\; t}} \cdot {X\left( {S,{t_{0} + {\Delta\; t}}} \right)}}{\mathbb{d}\left( {\ln\; S} \right)}}} \right\rbrack}$where PV[·] means “the present or discounted value of”.

By direct differentiation of the above formula, it can be proven thatthe “near-term vega,” (defined as the sensitivity of the value of theoption to the near-term volatility, σ₀), and the gamma are related bythe following equation:

$\frac{\partial X}{\partial\sigma_{0}} = {{2\sigma_{0}\frac{\partial X}{\partial\left( \sigma_{0}^{2} \right)}} = {\sigma_{0}S_{0}^{2}\frac{\partial^{2}X}{\partial S_{0}^{2}}\Delta\; t}}$

(In the case of any simple option whose payout is determined purely bythe value of the underlying at expiration, the above formula is correcteven when Δt is taken to be the entire time remaining to expiration ofthe option. This then provides a relationship between the full-term vegaand the gamma of these options).

Importantly, this equation demonstrates that the near-term vega and thegamma are in a common ratio for all options with the same underlyingwhich meet the criterion stated above.

If Δt refers to the remaining time until the Second Reference Time of anear-term inventive contract as described above, (i.e. θ), then it isclear that the vega and gamma of that contract are related by the aboveformula. This will also be true of any derivative meeting the criterionabove having an underlying which either is the same as that of thecontract or which may be expressed in terms of the underlying of thatcontract.

Because of the common ratio between the near-term vega and the gamma,the number of contracts which is required to hedge the gamma of optionpositions (determined as the ratio of the gamma of the option positionsto the gamma of one contract) is the same number that hedges thenear-term vega of that option portfolio (determined as the ratio of thenear-term vega of the option portfolio to the vega of one contract). Insummary, an appropriate number of contracts will simultaneously hedgeboth the gamma and the near-term vega. (The exposure to changes in moredistant volatility may be hedged with more distant contracts).

Even for options which do not meet the specified criterion, therelationship between the near-term vega and the gamma may be very nearlytrue. (For example, for an option which is contractually capable ofbeing exercised between t₀ and t₀+Δt prior to its expiration date, butwhere the conditions at t₀ dictate that exercise during that period isvery unlikely to occur).

Additional embodiments comprise option contracts which give the buyerthe right but not the obligation to buy or sell a contract in the formof one of the embodiments above at a set strike at some time or timesnot later than the First Reference Time. Because the strike willcorrespond to a particular level of volatility, these contracts areeffectively options on volatility.

For all the embodiments within the scope of this invention, it is notnecessary that the underlying for the contract exists at any particulartime other than the First and Second Reference Times. For example, whenthe underlying of the inventive contract is the price of a futurescontract, that futures contract may not yet have been made available fortrading at the time that the inventive contract is first made availablefor trading.

It should be understood that the invention also pertains to thesituation in which the contract specifies that different underlyings areobserved at the First and Second Reference Times if the level of oneunderlying at one of the Reference Times is related to the level of theother underlying. Examples of this are (i) the price of an equity indexfuture at the First Reference Time and the level of the relevant equityindex at the Second Reference Time, when the Second Reference Time isthe time at (or around) which the settlement price of the equity indexfuture is determined, and (ii) the yield corresponding to the price ofan interest rate future at the First Reference Time and the relevantinterest rate at the Second Reference Time, when the Second ReferenceTime is the time at (or around) which the settlement price of theinterest rate future is determined.

Also within the scope of this invention is a contract as describedherein which incorporates some averaging and/or rounding/truncation ofthe level of the underlying at (or around) the First and/or SecondReference Time, or of the strike(s).

Additionally within the scope of this invention is a contract whichbecomes, converts into or requires delivery of (from one party to theother) an option, straddle or strangle position with strike(s)determined with reference to the level of the underlying observed at oraround the First Reference Time in accordance with one of theaforementioned embodiments.

Looking at the drawings, FIG. 1 is an exemplary flow of the inventiveprocess. Contracts are made available for trading on an exchange, withcontract conditions specified. Levels of the underlying are determinedat first and second reference times. Then, any remaining contracts aresettled.

FIGS. 2 a and 2 b illustrate an exemplary flow of transactions relativeto option contracts traded on a futures and options exchange. By way ofexample, traders may initiate a sale or a purchase of option contracts,and otherwise may engage in trading of such contracts or other financialinstruments. Traders can be individual investors, investment firms ofvarying sizes, companies with pension plans under management, largefinancial institutions such as investment banks and/or governments, orany other type of individual or business entity which engages in tradingof financial instruments.

Referring to FIG. 2 a, in the normal course, each order (indicated bythe suffix 1 through to N), is communicated by a trader (100-1 to 100-N,not necessarily all distinct) to one of the trader's preferred brokers(150-1 to 150-N, not necessarily all distinct) through any number ofmedia 110-1 through 110-N. “Media” may include, but are not limited to,wide area networks, local area networks, the Internet, telephone orders,e-mail orders, face to face orders, “snail mail” service, hand delivery,or through other courier services. Some traders may have direct accessto the exchange and need not use a broker, (e.g. those known as “locals”who trade for their own account at an open-outcry exchange, and somelarge institutions who prefer to execute their own orders) in which casethe trader and the broker are coincident. The orders are submitted tothe exchange (200) by any number of media, (160-1 to 160-N), asdescribed above and possibly including open outcry, as permitted by theexchange in the hope of finding willing counterparties. Normally, thepriority of execution at any time amongst buy orders is on the qualityof the price (highest first) and then, (amongst buy orders at the sameprice), on a first-come-first-served basis. Similarly, the priority ofexecution at any time amongst sell orders is normally on the quality ofthe price (lowest first) and then, (amongst sell orders at the sameprice), on a first-come-first-served basis.

Looking now at FIG. 2 b, successfully executed orders (or parts thereof)are then reported back to the broker by the exchange (by any number ofmedia, 170-1 to 170-N, as described above), or the broker may experiencethe execution directly at the exchange in the case of open outcry. Thebroker reports the successfully executed orders back to the trader (byany number of media, 120-1 to 120-N, again as described above).Additionally the broker and/or the exchange (by any number of media,180-1 to 180-N and 210-1 to 210-N respectively, as described above) willreport the successfully executed order to the clearinghouse member,“CM,” (one of 250-1 to 250-N, not necessarily all distinct) with whomthe trader holds its account. The clearinghouse (300) matches buyers andsellers of the option contracts by acting as the seller to each buyerand the buyer to each seller. This approach ensures that parties are notsubject to default by their counterparties and eliminates the need tokeep track of the buyer and seller of each individual contract traded.The clearinghouse margins each of the clearinghouse members, and theclearinghouse members margin each of the traders who hold their accountwith them.

Some orders may not be executed in their entirety, because the order inrespect of unexecuted contracts is withdrawn by the trader or becausethere is insufficient demand for/supply of contracts at the priceproffered for sale/purchase.

It should not be inferred from these diagrams that the communicationsillustrated occur simultaneously. In reality, orders are submitted on alarge number of occasions throughout the period that the contract isavailable for trading. The execution of matching concurrent buy and sellorders on the exchange (up to the lower of the buy quantity and sellquantity) does occur at essentially the same time. The reporting of bothsides of the executed trades (to the traders and/or to the clearingmember/clearing house) may occur simultaneously depending on how thisinformation is reported and conveyed.

FIG. 3 a illustrates calculation, in the Black-Scholes framework and fora particular implied volatility, of the price of an option contract perunit of the contract's specified currency amount prior to the FirstReference Time in the case that the underlying is the price of a futurescontract or the corresponding yield of an interest rate futurescontract, and the option contract's payout is equal to the percentagegain, if any, (in the case of a call-type option) or the percentagefall, if any, (in the case of a put-type option) of the underlying overthe Reference Period multiplied by a specified currency amount. Sincethe price at this point is equal for both call and put options, the sameprocess can by used to find the prices of both. This value can beachieved by inputting the discount factor, (which may have beendetermined from interest rates and the time from the payment of thepremium until the final settlement of outstanding contracts), theimplied volatility of the underlying relating to the Reference Period,and the length of the Reference Period.

FIG. 3 b illustrates a calculation of the price of such a call-typeoption contract after the First Reference Time. Per the formulaediscussed above, inputs include the price of the underlying futurescontract or the corresponding yield of the underlying interest ratefutures contract, the strike price, the implied volatility relating tothe remaining time until the Second Reference Time, the discount factor,and the length of the time remaining in the Reference Period.

FIG. 3 c illustrates a calculation of the price of such a put-typeoption contract after the First Reference Time using the put-call parityrelationship. Input are the price for the same option if it were acall-type option, the strike price, the price of the underlying futurescontract or the corresponding yield of the underlying interest ratefutures contract, and the discount factor.

From the foregoing mathematics, it would be equally possible tocalculate V; using precisely the same inputs as for the call-typeoption. Equally, it is possible to calculate the price of the call-typeoption (through the put-call parity relationship) using the price of theput-type option, the strike price, the price of the underlying futurescontract or the corresponding yield of the underlying interest ratefutures contract, and the discount factor.

It also should be noted that it is possible to determine the impliedvolatility which, in accordance (for example) with the methodillustrated in FIG. 3 a, would generate a given price. This calculationmay employ a search using known techniques, including the Newton-Raphsonmethod.

While the present invention has been described in detail above withreference to several embodiments, various modifications and changeswithin the scope and spirit of the invention will be apparent to thoseof working skill in this technological field. Accordingly, the inventionis limited only by the scope of the following claims.

What is claimed is:
 1. A computer-implemented method for trading,between a buyer and a seller at an exchange, a futures exchange, anoptions exchange, or a futures and options exchange, the method beingperformed on a computer system, the method comprising: receiving, by thecomputer system, first inputs from the buyer and seller corresponding toa standardized form of contract for trading at a price on which thebuyer and the seller agree, the standardized form of contract havingcontract terms requiring the buyer and seller to settle based on a finalsettlement price; receiving, by the computer system, second inputs fromthe buyer and the seller at or prior to a first reference time whereinthe second inputs include bid prices and offer prices for thestandardized form of contract; and matching, in the computer system, thefirst inputs and the bid prices and the offer prices and forming acontract based, at least in part, on the matching; wherein said finalsettlement price is being determined, in accordance with the contractterms, by reference to both a first level of an underlying and a secondlevel of an underlying, the first level of the underlying beingsubstantially a value of the underlying at said first reference timeprior to settlement of the contract, the first reference time determinedin accordance with the contract terms, the underlying being a specifiedobservable quantity selected from a group consisting of a stock price, acommodity price, a financial asset price, a basket of financial assetsprice, a financial index value and a financial contract price, thesecond level of the underlying being substantially the value of theunderlying at a second reference time prior to settlement of thecontract, the second reference time determined in accordance with thecontract terms and being later than said first reference time; andwherein the contract is available to trade before said first referencetime.
 2. The computer-implemented method of claim 1, wherein the finalsettlement price is determined, in accordance with the contract terms,to be one of: i. substantially proportional to the excess of said secondlevel over a call strike level, if said second level is greater thansaid call strike level, and zero if said second level is less than orequal to said call strike level, wherein said call strike level issubstantially said first level subject to a multiplicative adjustment oradditive adjustment, or no adjustment; ii. substantially proportional tothe excess of a put strike level over said second level, if said secondlevel is less than said put strike level, and zero if said second levelis greater than or equal to said put strike level, wherein said putstrike level is substantially said first level subject to amultiplicative adjustment or additive adjustment, or no adjustment; andiii. substantially proportional to the excess of said second level overa call strike level, if said second level is greater than said callstrike level, substantially proportional to the excess of a put strikelevel over said second level, if said second level is less than said putstrike level, and zero if said second level is less than or equal tosaid call strike level and greater than or equal to said put strikelevel, wherein said call strike level is substantially said first levelsubject to a multiplicative adjustment or additive adjustment, or noadjustment, and wherein said put strike level is substantially saidfirst level subject to a multiplicative adjustment or additiveadjustment, or no adjustment.
 3. The computer-implemented method ofclaim 1, wherein the final settlement price is determined, in accordancewith the contract terms, to be one of: i. substantially proportional tothe square of the excess of said second level over a call strike level,if said second level is greater than said call strike level, and zero ifsaid second level is less than or equal to said call strike level,wherein said call strike level is substantially said first level subjectto a multiplicative adjustment or additive adjustment, or no adjustment;ii. substantially proportional to the square of the excess of a putstrike level over said second level, if said second level is less thansaid put strike level, and zero if said second level is greater than orequal to said put strike level, wherein said put strike level issubstantially said first level subject to a multiplicative adjustment oradditive adjustment, or no adjustment; and iii. substantiallyproportional to the square of the excess of said second level over acall strike level, if said second level is greater than said call strikelevel, substantially proportional to the square of the excess of a putstrike level over said second level, if said second level is less thansaid put strike level, and zero if said second level is less than orequal to said call strike level and greater than or equal to said putstrike level, wherein said call strike level is substantially said firstlevel subject to a multiplicative adjustment or additive adjustment, orno adjustment, and wherein said put strike level is substantially saidfirst level subject to a multiplicative adjustment or additiveadjustment, or no adjustment.
 4. The computer-implemented method ofclaim 1, wherein the final settlement price, in accordance with thecontract terms, is dependent on: the quotient of said second level ofthe underlying divided by said first level of the underlying; minus one,or another constant; minus the logarithm of the quotient of said secondlevel of the underlying divided by said first level of the underlying.5. The computer-implemented method of claim 1, wherein the contract istraded by telephone, internet, or a wide area network.
 6. Acomputer-implemented method for submitting trade order information to anexchange, a futures exchange, an options exchange, or a futures andoptions exchange, the method being performed on a computer system, themethod comprising: transmitting, by the computer system, first outputscorresponding to a standardized form of contract for trading at a priceon which a buyer and a seller agree, the standardized form of contracthaving contract terms requiring the buyer and seller to settle based ona final settlement price; and transmitting, by the computer system,second outputs at or prior to a first reference time wherein the secondoutputs include at least one bid price or offer price for thestandardized form of contract; wherein said final settlement price isbeing determined, in accordance with the contract terms, by reference toboth a first level of an underlying and a second level of an underlying,the first level of the underlying being substantially a value of theunderlying at said first reference time prior to settlement of thecontract, the first reference time determined in accordance with thecontract terms, the underlying being a specified observable quantityselected from a group consisting of a stock price, a commodity price, afinancial asset price, a basket of financial assets price, a financialindex value and a financial contract price, the second level of theunderlying being substantially the value of the underlying at a secondreference time prior to settlement of the contract, the second referencetime determined in accordance with the contract terms and being laterthan said first reference time; and wherein the contract is available totrade before said first reference time.
 7. The computer-implementedmethod of claim 6, wherein the final settlement price is determined, inaccordance with the contract terms, to be one of: i. substantiallyproportional to the excess of said second level over a call strikelevel, if said second level is greater than said call strike level, andzero if said second level is less than or equal to said call strikelevel, wherein said call strike level is substantially said first levelsubject to a multiplicative adjustment or additive adjustment, or noadjustment; ii. substantially proportional to the excess of a put strikelevel over said second level, if said second level is less than said putstrike level, and zero if said second level is greater than or equal tosaid put strike level, wherein said put strike level is substantiallysaid first level subject to a multiplicative adjustment or additiveadjustment, or no adjustment; and iii. substantially proportional to theexcess of said second level over a call strike level, if said secondlevel is greater than said call strike level, substantially proportionalto the excess of a put strike level over said second level, if saidsecond level is less than said put strike level, and zero if said secondlevel is less than or equal to said call strike level and greater thanor equal to said put strike level, wherein said call strike level issubstantially said first level subject to a multiplicative adjustment oradditive adjustment, or no adjustment, and wherein said put strike levelis substantially said first level subject to a multiplicative adjustmentor additive adjustment, or no adjustment.
 8. The computer-implementedmethod of claim 6, wherein the final settlement price is determined, inaccordance with the contract terms, to be one of: i. substantiallyproportional to the square of the excess of said second level over acall strike level, if said second level is greater than said call strikelevel, and zero if said second level is less than or equal to said callstrike level, wherein said call strike level is substantially said firstlevel subject to a multiplicative adjustment or additive adjustment, orno adjustment; ii. substantially proportional to the square of theexcess of a put strike level over said second level, if said secondlevel is less than said put strike level, and zero if said second levelis greater than or equal to said put strike level, wherein said putstrike level is substantially said first level subject to amultiplicative adjustment or additive adjustment, or no adjustment; andiii. substantially proportional to the square of the excess of saidsecond level over a call strike level, if said second level is greaterthan said call strike level, substantially proportional to the square ofthe excess of a put strike level over said second level, if said secondlevel is less than said put strike level, and zero if said second levelis less than or equal to said call strike level and greater than orequal to said put strike level, wherein said call strike level issubstantially said first level subject to a multiplicative adjustment oradditive adjustment, or no adjustment, and wherein said put strike levelis substantially said first level subject to a multiplicative adjustmentor additive adjustment, or no adjustment.
 9. The computer implementedmethod of claim 6, wherein the final settlement price, in accordancewith the contract terms, is dependent on: the quotient of said secondlevel of the underlying divided by said first level of the underlying;minus one, or another constant; minus the logarithm of the quotient ofsaid second level of the underlying divided by said first level of theunderlying.
 10. The computer-implemented method of claim 6, wherein thetrade order information is submitted by telephone, internet, or a widearea network.
 11. A computer-implemented method for clearing contractsthorough a clearinghouse, the method being performed on a computersystem, the method comprising: receiving, by the computer system, firstinputs corresponding to a standardized form of contract for trading at aprice on which a buyer and a seller agree, the standardized form ofcontract having contract terms requiring the buyer and seller to settlebased on a final settlement price; receiving, by the computer system,second inputs at or prior to a first reference time corresponding to theidentity of a buyer (or their agent) or a seller (or their agent); andreceiving, by the computer system, third inputs at or prior to a firstreference time wherein the second inputs include a price for thestandardized form of contract; wherein said final settlement price isbeing determined, in accordance with the contract terms, by reference toboth a first level of an underlying and a second level of an underlying,the first level of the underlying being substantially a value of theunderlying at said first reference time prior to settlement of thecontract, the first reference time determined in accordance with thecontract terms, the underlying being a specified observable quantityselected from a group consisting of a stock price, a commodity price, afinancial asset price, a basket of financial assets price, a financialindex value and a financial contract price, the second level of theunderlying being substantially the value of the underlying at a secondreference time prior to settlement of the contract, the second referencetime determined in accordance with the contract terms and being laterthan said first reference time, and wherein the contract is available totrade before said first reference time.
 12. The computer-implementedmethod of claim 11, wherein the final settlement price is determined, inaccordance with the contract terms, to be one of: i. substantiallyproportional to the excess of said second level over a call strikelevel, if said second level is greater than said call strike level, andzero if said second level is less than or equal to said call strikelevel, wherein said call strike level is substantially said first levelsubject to a multiplicative adjustment or additive adjustment, or noadjustment; ii. substantially proportional to the excess of a put strikelevel over said second level, if said second level is less than said putstrike level, and zero if said second level is greater than or equal tosaid put strike level, wherein said put strike level is substantiallysaid first level subject to a multiplicative adjustment or additiveadjustment, or no adjustment; and iii. substantially proportional to theexcess of said second level over a call strike level, if said secondlevel is greater than said call strike level, substantially proportionalto the excess of a put strike level over said second level, if saidsecond level is less than said put strike level, and zero if said secondlevel is less than or equal to said call strike level and greater thanor equal to said put strike level, wherein said call strike level issubstantially said first level subject to a multiplicative adjustment oradditive adjustment, or no adjustment, and wherein said put strike levelis substantially said first level subject to a multiplicative adjustmentor additive adjustment, or no adjustment.
 13. The computer-implementedmethod of claim 11, wherein the final settlement price is determined, inaccordance with the contract terms, to be one of: i. substantiallyproportional to the square of the excess of said second level over acall strike level, if said second level is greater than said call strikelevel, and zero if said second level is less than or equal to said callstrike level, wherein said call strike level is substantially said firstlevel subject to a multiplicative adjustment or additive adjustment, orno adjustment; ii. substantially proportional to the square of theexcess of a put strike level over said second level, if said secondlevel is less than said put strike level, and zero if said second levelis greater than or equal to said put strike level, wherein said putstrike level is substantially said first level subject to amultiplicative adjustment or additive adjustment, or no adjustment; andiii. substantially proportional to the square of the excess of saidsecond level over a call strike level, if said second level is greaterthan said call strike level, substantially proportional to the square ofthe excess of a put strike level over said second level, if said secondlevel is less than said put strike level, and zero if said second levelis less than or equal to said call strike level and greater than orequal to said put strike level, wherein said call strike level issubstantially said first level subject to a multiplicative adjustment oradditive adjustment, or no adjustment, and wherein said put strike levelis substantially said first level subject to a multiplicative adjustmentor additive adjustment, or no adjustment.
 14. The computer implementedmethod of claim 11, wherein the final settlement price, in accordancewith the contract terms, is dependent on: the quotient of said secondlevel of the underlying divided by said first level of the underlying;minus one, or another constant; minus the logarithm of the quotient ofsaid second level of the underlying divided by said first level of theunderlying.
 15. The computer-implemented method of claim 11, wherein theinputs are received by telephone, internet, or a wide area network. 16.A computer-implemented method for displaying trade order information,reflecting buying and selling interest received by an exchange, afutures exchange, an options exchange, a futures and options exchange,or a clearinghouse, the method being performed on a computer system, themethod comprising: displaying, by the computer system, first outputscorresponding to a standardized form of contract for trading at a priceon which a buyer and a seller agree, the standardized form of contracthaving contract terms requiring the buyer and seller to settle based ona final settlement price; and displaying, by the computer system, secondoutputs at or prior to a first reference time wherein the second outputsinclude at least one bid price or offer price for the standardized formof contract; wherein said final settlement price is being determined, inaccordance with the contract terms, by reference to both a first levelof an underlying and a second level of an underlying, the first level ofthe underlying being substantially the value of the underlying at saidfirst reference time prior to settlement of the contract, the firstreference time determined in accordance with the contract terms, theunderlying being a specified observable quantity selected from a groupconsisting of a stock price, a commodity price, a financial asset price,a basket of financial assets price, a financial index value and afinancial contract price, the second level of the underlying beingsubstantially the value of the underlying at a second reference timeprior to settlement of the contract, the second reference timedetermined in accordance with the contract terms and being later thansaid first reference time, and wherein the contract is available totrade before said first reference time.